Question: A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is  $r$?
Explanation: Let the dimensions of $P$ be $x$, $y$, and $z$. The sum of the lengths of the edges of $P$ is $4(x+y+z)$, and the surface area of $P$ is $2xy+2yz+2xz$, so \[
x+y+z=28 \quad\text{and}\quad 2xy+2yz+2xz=384.
\] Each internal diagonal of $P$ is a diameter of the sphere, so \begin{align*}
(2r)^2&=(x^2+y^2+z^2)\\
&=(x+y+z)^2-(2xy+2xz+2yz) \\
&= 28^2-384\\& = 400.
\end{align*} So $2r = 20$ and  $r=\boxed{10}$.

Note: There are infinitely many positive solutions of the system $x+y+z=28$, $2xy+2yz+2xz=384$, so there are infinitely many non-congruent boxes meeting the given conditions, but each can be inscribed in a sphere of radius 10.